Question

Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to two decimal places where appropriate.)

$\sin \left(\theta \right)=\frac{\sqrt{3}}{2}$

Answer 1

Step-1: Given Equation

$\sin \left(\theta \right)=\frac{\sqrt{3}}{2}$

Step-2: Finding the values of $\theta$.

Solving that equation is finding the value of the angle, $\theta$, whose sine is $\frac{\sqrt{3}}{2}$.

The function that returns the value of the angle whose sine is $\frac{\sqrt{3}}{2}$ is the inverse function of the sine and it has a special name: arc sine or arcsine.

Hence, $\theta ={\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Using the knowledge of the notable angles to solve that equation.

The function sine is positive in the first and second quadrants.

The sine of $60°$ is $\frac{\sqrt{3}}{2}$ ,

Hence the first value of $\theta$is $60°$

$60°$is the reference angle in the second quadrant, and the angle searched is $180°-$ reference angle.

$\Rightarrow 180°-60°=120°$

So, $120°$ is the other solution of the equation.

Hence, $\theta =60°,120°$.